# Remarks on the Preservation of the Almost Fixed Point Property Involving Several Types of Digitizations

## Abstract

**:**

## 1. Introduction

## 2. Several Kinds of Digital Topological Categories, DTC, KTC and MTC

**Definition**

**1.**

**Proposition**

**1**

**.**Let $(X,{k}_{0})$ and $(Y,{k}_{1})$ be digital images in ${\mathbb{Z}}^{{n}_{0}}$ and ${\mathbb{Z}}^{{n}_{1}}$, respectively. A function $f:(X,{k}_{0})\to (Y,{k}_{1})$ is $({k}_{0},{k}_{1})$-continuous if and only if for every $x\in X$, $f({N}_{{k}_{0}}(x,1))\subset {N}_{{k}_{1}}(f(x),1)$.

- The set of objects $(X,k)$, denoted by $Ob(DTC)$;
- For every ordered pair of objects $(X,{k}_{1})$ and $(Y,{k}_{2})$, the set of all $({k}_{1},{k}_{2})$-continuous maps $f:(X,{k}_{1})\to (Y,{k}_{2})$ as morphisms.

**Proposition**

**2**

**.**Consider a bounded digital plane (or finite digital picture) $(X,k),k\in \{4,8\}$, i.e., $({[a,b]}_{\mathbb{Z}}\times {[c,d]}_{\mathbb{Z}}:=X,k)$.

**Theorem**

**1.**

**Proof.**

**Remark**

**1.**

## 3. Some Properties of a K-, an M-, a U- or an L-Digitization

**Definition**

**2**

**.**In ${\mathbb{R}}^{n}$, for each point $p:={({p}_{i})}_{i\in {[1,n]}_{\mathbb{Z}}}\in {\mathbb{Z}}^{n}$, we define the set ${N}_{K}(p):=\left\{{({x}_{i})}_{i\in {[1,n]}_{\mathbb{Z}}}\right\}$, which is called the local K-neighborhood of p associated with $({\mathbb{Z}}^{n},{\kappa}^{n})$, where $t\in \mathbb{Z}$ and

**Remark**

**2.**

**Definition**

**3**

**.**For a nonempty space $(X,{E}_{X}^{n})$, we define a K-digitization of X, denoted by ${D}_{K}(X)$, to be the space with K-topology

**Definition**

**4**

**.**In ${\mathbb{R}}^{2}$, for a point $p:=({p}_{1},{p}_{2})\in {\mathbb{Z}}^{2}$, we define the following neighborhood of p:

**Remark**

**3.**

**Definition**

**5**

**Remark**

**4.**

**Proposition**

**3.**

**Definition**

**6**

**.**Under $({\mathbb{R}}^{n},{E}_{U}^{n})$, for a point $p:={({p}_{i})}_{i\in {[1,n]}_{\mathbb{Z}}}\in {\mathbb{Z}}^{n}$, we define ${N}_{U}(p):=\left\{{({x}_{i})}_{i\in {[1,n]}_{\mathbb{Z}}}\right|{x}_{i}\in ({p}_{i}-\frac{1}{2},{p}_{i}+\frac{1}{2}]\}$, and we call ${N}_{U}(p)$ the U-localized neighborhood of p associated with $({\mathbb{R}}^{n},{E}_{U}^{n})$.

**Definition**

**7**

**.**Let ${D}_{U(k)}:({\mathbb{R}}^{n},{E}^{n})\to ({\mathbb{Z}}^{n},k)$ be the map defined by ${D}_{U(k)}(x)=p$, where $x\in {N}_{U}(p),\phantom{\rule{0.166667em}{0ex}}p\in {\mathbb{Z}}^{n}$ and the k-adjacency is taken according to the situation. Then, we say that ${D}_{U(k)}$ is a $U(k)$-digitization operator.

**Definition**

**8**

**.**Under $({\mathbb{R}}^{n},{E}_{L}^{n})$, for a point $p:={({p}_{i})}_{i\in {[1,n]}_{\mathbb{Z}}}\in {\mathbb{Z}}^{n}$, we define ${N}_{L}(p):=\{{({x}_{i})}_{i\in {[1,n]}_{\mathbb{Z}}}|{x}_{i}\in [{p}_{i}-\frac{1}{2},{p}_{i}+\frac{1}{2})\}$. We call ${N}_{L}(p)$ the L-localized neighborhood of p associated with $({\mathbb{R}}^{n},{E}_{L}^{n})$.

**Definition**

**9**

**.**Let ${D}_{L(k)}:({\mathbb{R}}^{n},{E}^{n})\to ({\mathbb{Z}}^{n},k)$ be the map defined by ${D}_{L(k)}(x)=p$, where $x\in {N}_{L}(p),\phantom{\rule{0.166667em}{0ex}}p\in {\mathbb{Z}}^{n}$ and the k-adjacency determined according to the situation. Then, we say that ${D}_{L(k)}$ is an $L(k)$-digitization operator.

**Definition**

**10**

**.**Let X be a subspace in $({\mathbb{R}}^{n},{E}_{U}^{n})$ (resp. $({\mathbb{R}}^{n},{E}_{L}^{n})$). The U- (resp. L-) digitization of X, denoted by ${D}_{U}(X)$ (resp. ${D}_{L}(X)$), is defined as follows:

**Proposition**

**4.**

## 4. Explorations of the Preservation of the AFPP of a Compact Plane into theAFPP of a K-, an M-, a U(k)-, or an L(k)-Digitized Space

**Question****1**- Let X be the set ${\prod}_{i\in \{1,2,\cdots ,n\}}{[-{l}_{i},{l}_{i}]}_{\mathbb{Z}}$. How about the FPP or the AFPP of the K-topological space $(X,{\kappa}_{X}^{n})$?
**Question****2**- Let Y be the set ${\prod}_{i\in \{1,2\}}{[-{l}_{i},{l}_{i}]}_{\mathbb{Z}}$. What about the AFPP of the M-topological space $(Y,{\gamma}_{Y})$?
**Question****3**- How about the preservation of the AFPP of a compact n-dimensional Euclidean cube into the AFPP of its $U(k)$-, or $L(k)$-digitized space?

**Lemma**

**1.**

**Proof.**

- Case 1
- Consider $U(p)$, where $p\in \{(2m,2n),(2m+1,2n+1)|m,n\in \mathbb{Z}\}$. Then, assume any M-continuous self-map f of $(U(p),{\gamma}_{U(p)})$. If p is mapped by f onto a point $q\in U(p)\backslash \left\{p\right\}$, then the map should be a constant map with $f(U(p))=\left\{q\right\}$ according to the M-continuity of f, which implies that $(U(p),{\gamma}_{U(p)})$ has the FPP with a fixed point q associated with the map f. In addition, in case $f(p)=p$, the assertion is trivial.
- Case 2
- Assume that $U(p)$ is a singleton. Then, it is obvious that $(U(p),{\gamma}_{U(p)})$ has the FPP.

**Proposition**

**5**

**Corollary**

**1.**

**Proof.**

**Definition**

**11**

**.**In MTC, we say that an M-continuous map $r:({X}^{\prime},{\gamma}_{{X}^{\prime}})\to (X,{\gamma}_{X})$ is an M-retraction if

- (1)
- $(X,{\gamma}_{X})$ is a subspace of $({X}^{\prime},{\gamma}_{{X}^{\prime}}),$ and
- (2)
- $r(a)=a$ for all $a\in (X,{\gamma}_{X})$.

**Lemma**

**2**

**.**For $(X,{\gamma}_{X})$ let $(A,{\gamma}_{A})$ be an M-retract of $(X,{\gamma}_{X})$. If $(X,{\gamma}_{X})$ has the AFPP, then $(A,{\gamma}_{A})$ also has the AFPP.

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

- (1)
- The functor ${D}_{M}$ does not preserve the AFPP,
- (2)
- The functor ${D}_{U(k)}$ preserves the AFPP if $k=8$,
- (3)
- The functor ${D}_{L(k)}$ preserves the AFPP if $k=8$

- (4)
- The functor ${D}_{U(k)}$ preserves the AFPP if $k={3}^{n}-1$,
- (5)
- The functor ${D}_{L(k)}$ preserves the AFPP if $k={3}^{n}-1$.

**Proof.**

- (1)
- For $(X,{E}_{X}^{2})(\subset ({\mathbb{R}}^{2},{E}^{2})$, since ${D}_{M}(X)$ is also M-connected [13] and furthermore that $({D}_{M}(X),{\gamma}_{{D}_{M}(X)})$ is a compact M-topological plane, by Theorem 2, we obtain that $({D}_{M}(X),{\gamma}_{{D}_{M}(X)})$ does not have the AFPP, which completes the proof.
- (2)
- Using Propositions 2 and 4, the proof is completed.
- (3)
- Using the method similar to the proof (2), we complete the proof.
- (4)
- For $(X:={[-1,1]}^{n},{E}_{X}^{n})(\subset ({\mathbb{R}}^{n},{E}^{n})$, it is obvious that $({D}_{U(k)}(X),k)$ is k-connected, $k={3}^{n}-1$. Hence, by Theorem 1, the digital image $({D}_{U(k)}(X),k),k={3}^{n}-1$ has the AFPP. Hence, ${D}_{U(k)}$ preserves the AFPP if $k={3}^{n}-1$.

- (5)
- It is obvious that $({D}_{L(k)}(X),k)$ is k-connected, $k={3}^{n}-1$. Hence, by Theorem 1, the digital image $({D}_{L(k)}(X),k),k={3}^{n}-1$ has the AFPP.

**Remark**

**5.**

## 5. Conclusions

## Funding

## Conflicts of Interest

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**Figure 1.**The non-AFPP of the digital 2-cube with 4-adjacency, $({[-1,1]}_{\mathbb{Z}}^{2}:=X,4)$. (1) Configuration of the map ${f}_{1}$; (2) Explanation of the map ${f}_{2}$.

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Han, S.-E. Remarks on the Preservation of the Almost Fixed Point Property Involving Several Types of Digitizations. *Mathematics* **2019**, *7*, 954.
https://doi.org/10.3390/math7100954

**AMA Style**

Han S-E. Remarks on the Preservation of the Almost Fixed Point Property Involving Several Types of Digitizations. *Mathematics*. 2019; 7(10):954.
https://doi.org/10.3390/math7100954

**Chicago/Turabian Style**

Han, Sang-Eon. 2019. "Remarks on the Preservation of the Almost Fixed Point Property Involving Several Types of Digitizations" *Mathematics* 7, no. 10: 954.
https://doi.org/10.3390/math7100954